Gaussian Process Computations¶
The main interface to the celerite solver is through the GP
class.
This main purpose of this class is to exposes methods for efficiently
computing the marginalized likelihood of a Gaussian Process (GP) model.
The covariance matrix for the GP will be specified by a kernel function as
described in the Kernel Building section.
The GP
class implements the Python: Modeling with submodels called
kernel
, mean
, and log_white_noise
so the modeling language can be
used to fit for parameters for each of these models.
Below, the methods of the GP
object are described in detail but the
most important methods to look at are probably GP.compute()
,
GP.log_likelihood()
, and GP.predict()
.

class
celerite.
GP
(kernel, mean=0.0, fit_mean=False, log_white_noise=inf, fit_white_noise=False, method=None)¶ The main interface to the celerite Gaussian Process solver
Parameters:  kernel – An instance of a subclass of
terms.Term
.  mean (Optional) – A simple mean value for the process. This can either
be a
float
or a subclass ofmodeling.Model
. (default:0.0
)  fit_mean (optional) – If
False
, all of the parameters ofmean
will be frozen. Otherwise, the parameter states are unaffected. (default:False
)  log_white_noise (Optional) – A white noise model for the process. The
exp
of this will be added to the diagonal of the matrix inGP.compute()
. In other words this model should be for the log of the _variance_ of the white noise. This can either be afloat
or a subclass ofmodeling.Model
. (default:inf
)  fit_white_noise (Optional) – If
False
, all of the parameters oflog_white_noise
will be frozen. Otherwise, the parameter states are unaffected. (default:False
)  method – Select a matrix solver method by name. This can be one of
(a)
simple
: a simple banded Gaussian elimination based method, (b)lapack
: an optimized band solver if compiled with LAPACK, (c)sparse
: a sparse solver if compiled with Eigen/Sparse, or (d)default
: uses heuristics to select the fastest method that is available.

apply_inverse
(y)¶ Apply the inverse of the covariance matrix to a vector or matrix
Solve
K.x = y
forx
whereK
is the covariance matrix of the GP with the white noise andyerr
components included on the diagonal.Parameters: Returns: The solution to the linear system. This will have the same shape as
y
.Return type: Raises: ValueError
– For mismatched dimensions.

compute
(t, yerr=1.123e12, check_sorted=True)¶ Compute the extended form of the covariance matrix and factorize
Parameters:  x (array[n]) – The independent coordinates of the data points. This array must be _sorted_ in ascending order.
 yerr (Optional[float or array[n]]) – The measurement uncertainties
for the data points at coordinates
x
. These values will be added in quadrature to the diagonal of the covariance matrix. (default:1.123e12
)  check_sorted (bool) – If
True
,x
will be checked to make sure that it is properly sorted. IfFalse
, the coordinates will be assumed to be in the correct order.
Raises: ValueError
– For unsorted data or mismatched dimensions.

dot
(y, kernel=None)¶ Dot the covariance matrix into a vector or matrix
Compute
K.y
whereK
is the covariance matrix of the GP without the white noise oryerr
values on the diagonal.Parameters:  y (array[n] or array[n, nrhs]) – The vector or matrix
y
described above.  kernel (Optional[terms.Term]) – A different kernel can optionally
be provided to compute the matrix
K
from a different kernel than thekernel
property on this object.
Returns: The dot product
K.y
as described above. This will have the same shape asy
.Return type: Raises: ValueError
– For mismatched dimensions. y (array[n] or array[n, nrhs]) – The vector or matrix

get_matrix
(x1=None, x2=None, include_diagonal=None)¶ Get the covariance matrix at given independent coordinates
Parameters:  x1 (Optional[array[n1]]) – The first set of independent coordinates.
If this is omitted,
x1
will be assumed to be equal tox
from a previous call toGP.compute()
.  x2 (Optional[array[n2]]) – The second set of independent
coordinates. If this is omitted,
x2
will be assumed to bex1
.  include_diagonal (Optional[bool]) – Should the white noise and
yerr
terms be included on the diagonal? (default:False
)
 x1 (Optional[array[n1]]) – The first set of independent coordinates.
If this is omitted,

log_likelihood
(y, _const=1.8378770664093453)¶ Compute the marginalized likelihood of the GP model
The factorized matrix from the previous call to
GP.compute()
is used socompute
must be called first.Parameters: y (array[n]) – The observations at coordinates x
fromGP.compute()
.Returns: The marginalized likelihood of the GP model. Return type: float Raises: ValueError
– For mismatched dimensions.

log_white_noise
¶ The white noise
modeling.Model

mean
¶ The mean
modeling.Model

predict
(y, t=None, return_cov=True, return_var=False)¶ Compute the conditional predictive distribution of the model
You must call
GP.compute()
before this method.Parameters: y (array[n]) – The observations at coordinates x
fromGP.compute()
. t (Optional[array[ntest]]): The independent coordinates where the
 prediction should be made. If this is omitted the coordinates will
be assumed to be
x
fromGP.compute()
and an efficient method will be used to compute the prediction.  return_cov (Optional[bool]): If
True
, the full covariance matrix  is computed and returned. Otherwise, only the mean prediction is
computed. (default:
True
)  return_var (Optional[bool]): If
True
, only return the diagonal of  the predictive covariance; this will be faster to compute than the
full covariance matrix. This overrides
return_cov
so, if both are set toTrue
, only the diagonal is computed. (default:False
)
Returns: mu
,(mu, cov)
, or(mu, var)
depending on the values ofreturn_cov
andreturn_var
. These output values are: (a) mu(ntest,)
: mean of the predictive distribution, (b) cov(ntest, ntest)
: the predictive covariance matrix, and (c) var(ntest,)
: the diagonal elements ofcov
.Raises: ValueError
– For mismatched dimensions.

sample
(x=None, diag=None, include_diagonal=False, size=None)¶ Sample from the prior distribution over datasets
Parameters:  x (Optional[array[n]]) – The independent coordinates where the
observations should be made. If
None
, the coordinates used in the last call tocompute
will be used.  diag (Optional[array[n] or float]) – If provided, this will be added to the diagonal of the covariance matrix.
 include_diagonal (Optional[bool]) – Should the white noise and/or
yerr
terms be included on the diagonal? (default:False
)  size (Optional[int]) – The number of samples to draw.
Returns: The samples from the prior distribution over datasets.
Return type:  x (Optional[array[n]]) – The independent coordinates where the
observations should be made. If
 kernel – An instance of a subclass of